Elliptic curves over finite fields have been used in recent public key clyptosysterns and authentication. The discrete logarithm problems over the elliptic curves can resist all known subexponential attacks which then can implement cryptographic schemes in higher speed and less key sizes while retain the same security comparing with traditional cryptographic functions. In this research, we propose efficient algorithms to construct secure elliptic arid hyperelliptic cryptosysterns.The point-counting algorithms to construct explicitly secure elliptic curves for cryptosystems can find secure curves over finite fields from randomly selected elliptic curves, but are quite time consuming especially when one wishes to choose different curves for different users or periodically change curves over finite fields in the same cryptosystem, Elliptic curves over number fields with CM can be used to design non-isogenous elliptic cryptosystems over finite fields efficiently. The existing algorithm to
… Morebuild such CM curves, costing exponential time of computations OMICRON(2^<5h/2>h^<21/4>) where h is the class number of the endomorphism ring of the CM curve. Thus it carl only be used to construct CM elliptic curves with small class numbers.We propose polynomial time algorithms in h to build CM elliptic curves over number fields : by lifting the ring class equations from small finite fields thus constructing CM curves. Its complexity is shown as in a polynomial time in h, i.e., . OMICRON(h^7). Furthermore, these algorithms are also extented to hyperelliptic cryptosystems, for which no efficient algorithm is known until now for construction of secure hyperelliptic curves. We propose efficient algorithms to construct secure discrete logarithm problems over hyperelliptic curves based on Weil elements. The lifting approach to build CM curves is also generalized to Jacobian varieties of algebraic curves of higher genera.従来の虚数乗法を持つ楕円曲線の構成法は、保形関数の級数展開を用いているため、その計算量は、虚数乗法体の類数の指数時間を演算が必要である。本研究では、虚数乗法を持つ楕円曲線の還元の性質を利用して、小さな有限体上の楕円曲線を持ち上げることに成功して、虚数乗法体の類数の多項式時間の高速アルゴリズムを提案している。具体的にソフトウェア実現に有利な大きな素体において、そして、ハードウェア実現しやすい標数2の拡大体上において、ShanksのBaby-step-giant-step攻撃法及びMOVリダクション攻撃に安全な楕円暗号系を構成している。さらに、超楕円暗号系においては、安全な超楕円曲線の構成は、楕円曲線に較べて格段に難しく、現在現実的に曲線を構成する方法は知られていない。本研究では、アーベル多様体のCM理論を利用することで、虚数乗法を持つ超楕円或は代数曲線を利用して、安全な暗号系の効率的な構成法を提案している。さらに、上記持ち上げによる楕円曲線の構成法を超楕円曲線へも拡張している。 Less